"At the end of the 20th century, relations between number theory and physics was considered by Keating-Snaith and Katz-Sarnak, respectively. Keating and Snaith conjectured a similarity between moments of $L$-functions and those of characteristic polynomials of random matrices. Later, Katz and Sarnak suggested that zeros of $L$-functions should distribute like eigenvalues of random matrices. In this talk, we focus on zeros of $L$-functions and suggest the weighted density conjecture of zeros for families of $L$-functions. Moreover we show two pieces of evidence supporting this conjecture. The first evidence is on symmetric power $L$-functions attached to modular forms,and the second one is on Dirichlet $L$-functions. The second result is a joint work with Ade Irma Suriajaya (Kyushu University)."